### Stocks are not Well Behaved

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So I stole this idea from Benoit Mandelbrot from his book *The Misbehavior of Markets* (which is an interesting book btw). I looked at daily % change in the price of the S&P 500 going back to 1950 (which is where yahoo finance's data begins). Here's what it looks like:

So the question to ask this: what would it look like if the volatility of day-to-day % change price movements look like if the returns were normally distributed. To do this, I used Random.org's Gaussian Random Number Generator to generate a list of random numbers comparable to the distribution for S&P 500 % price change. What I get looks like this:

Some general statistics:

Both had 15715 observations.

The Mean was 0.03% for the S&P 500 and 0.04% for the Normal Distribution Data.

Both had Standard Deviation of 0.98%

The Max/Min for the Normal Distribution Data were 3.00%/-2.90%.

The Max/Min for the S&P 500 Data were 11.58%/ -20.47%

As expected, the normal distribution stayed within about 3 standard deviations of the mean which should account for about 99.7% of the data. Contrast that with the S&P 500 daily changes and we find many points going outside of 3 standard deviations.

The fact that the volatility for stocks is not normally distributed has important implications. As an example, the Black-Scholes model for option pricing assumes that volatility follows a normal distribution. That's why option implied volatility shows a volatility smile, accounting for the fatter tails.

## 4 Comments – Post Your Own

#1)On June 17, 2012 at 10:52 PM,HarryCaraysGhost (88.00)wrote:What would the chart look like if you included dividends and dollar cost averaging?

Report this comment#2)On June 17, 2012 at 11:35 PM,portefeuille (98.74)wrote:a figure showing daily changes of the S&P500 index.

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a figure showing daily changes of the DAX starting on November 27, 1990.

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also see this post.

July 16, 2010, Roller Coaster

Report this comment#3)On June 17, 2012 at 11:38 PM,portefeuille (98.74)wrote:

Report this comment#4)On June 18, 2012 at 1:43 AM,somrh (87.87)wrote:Harry, I have no idea. I'm not even sure what it would mean to add those in. For example, on which days would you add dividends to the S&P 500? Would you take the quarterly SPY distributions? Or would you figure out for each stock in the index when it pays a dividend. Then how would you add them in?

portefeuille, nice video. And I think your charts look prettier. DAX looks a bit more volatile. I wonder how much that has to do with the fact that DAX index has only 30 stocks.

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